Informal insurance in social networks

Links can be broken, if people have “betrayed”their relationship with an unkept promise to insure each other, but new links cannot be formed.While there are merits to an alternative exer

New York University and Instituto de An´alisis Econ´omico (CSIC)

July 2005, revised September 2006

AbstractThis paper studies bilateral insurance schemes across networks of individuals While transfersare based on social norms, each individual must have the incentive to abide by those norms,and so we investigate the structure of self-enforcing insurance networks Network links play twodistinct and possibly conflictual roles First, they act as conduits for transfers Second, they act

as conduits for information These features affect the scope for insurance, as well as the severity

of punishments in the event of noncompliance Their interaction leads to a characterization ofstable networks as networks which are suitably “sparse”, the degree of sparseness being related

to the length of the minimal cycle that connects any triple of agents As corollaries, we find thatboth “thickly connected” networks (such as the complete graph) and “thinly connected” networks(such as trees) are likely to be stable, whereas intermediate degrees of connectedness jeopardizestability Finally, we study in more detail the notion of networks as conduits for transfers, bysimply assuming a punishment structure (such as autarky) that is independent of the precise

architecture of the network This allows us to isolate a bottleneck effect: the presence of certain

key agents who act as bridges for several transfers Bottlenecks are captured well in a feature of

trees that we call decomposability, and we show that all decomposable networks have the same

stability properties and that these are the least likely to be stable

JEL Classification Numbers: D85, D80, 012, Z13

Keywords: social networks, reciprocity networks, norms, informal insurance.

An earlier draft was written while Genicot and Ray were visiting the London School of ics; we thank the LSE for their hospitality Ray is grateful for funding from the National ScienceFoundation under grant no 0241070 We are grateful for comments by seminar participants atBarcelona, Cergy, CORE, Essex, Helsinki LSE/UCL, Oxford and University of Marylandand andaudiences at the NEUDC in Montr´eal, the AEA Meetings in Philadelphia, the SED in Budapest,conferences on networks and coalitions in Vaxholm and Guanajato, and the Conference in trib-ute to Jean-Jacques Laffont in Toulouse Contact the authors at francis.bloch@univmed.fr,gg58@georgetown.edu, and debraj.ray@nyu.edu

Econom-1 IntroductionThis paper studies networks of informal insurance Such networks exist everywhere, but espe-cially so in developing countries and in rural areas where credit and insurance markets are scarceand income fluctuations are endemic Yet it is also true that everybody does not enter into recip-rocal insurance arrangements with everybody else, even in relatively small village communities.

A recent empirical literature (see for instance Fafchamps (1992), Fafchamps and Lund (2003),and Murgai, Winters, DeJanvry and Sadoulet (2002)) shows that insurance schemes often takesplace within subgroups in a community One obvious reason for this is that everyone may notknow one another at a level where such transactions become feasible A community — based

on friends, extended family, kin or occupation — comes first, insurance comes later This is thestarting point of our paper.1

Once this view is adopted, however, it is clear that an “insurance community” is not a closed

multilateral grouping A may insure with B, and B with C, but A and C may have nothing to do with each other The appropriate concept, then, is one of an insurance network Empirically, such

networks have attracted attention and have recently been mapped to some extent: for instance,Stack (1974), Wellman (1992), de Weerdt (2002), Dercon and de Weerdt (2000), and Fafchampand Gubert (2004) reveal a complex architecture of risk-sharing networks

The very idea of an insurance network rather than a group suggests that our existing notion

of insurance as taking place within an multilateral “club” of several people may be misleading

Of course, such clubs may well exist, but a significant segment of informal insurance transactions

is bilateral A and B will have their very own history of kindness, reciprocity or betrayal.

In these histories, either party may have been have cognizant of (and taken into account) herpartner’s obligations to (or receipts from) a third individual, but the fundamental relationship isnevertheless bilateral

A principal aim of our paper is to build a model of risk-sharing networks which captures thisfeature A prior relationship is used to define the network Once in place, both insurance and thetransfer of information is limited by this network Links can be broken, if people have “betrayed”their relationship with an unkept promise to insure each other, but new links cannot be formed.While there are merits to an alternative exercise in which new links can be deliberately formed,there is also much to be gained from studying a view in which existing links are nonstrategic,and so we view these two frameworks as complementary.2

In the model studied here, only “directly linked” agents in some given network make transfers

to each other, though they are aware of the (aggregate) transfers each makes to others We view

insurance as being based on internalized norms regarding mutual help A bilateral insurance

norm between two linked agents specifies consumptions for every linked pair of individuals, as

a function of various observables such as their identities, the network component they belong

to, their income realizations and the transfers made to or received from other agents Thesetransfers are taken as given by the linked pair, but are obviously endogenous for society as

1 As Genicot and Ray (2003, 2005) have argued, there may also be strategic reasons for limited group formation Genicot and Ray build on a large literature which studies insurance schemes with self-enforcement constraints; see, e.g., Posner (1980), Kimball (1988), Coate and Ravallion (1993), Kocherlakota (1996), Kletzer and Wright (2000) and Ligon, Thomas and Worrall (2002)).

2 In research that has recently come to our attention, Bramoull´e and Kranton (2005) study the formation of insurance networks under the assumption of equal division and perfect enforcement By contrast, our paper studies

a family of insurance schemes — including equal division — in an explicit context of self-enforcement, but assumes that the network is given for exogenous reasons such as friendship, family, or social contacts In our model, links can be broken but new links can never be formed.

a whole We therefore introduce the notion of a consistent consumption allocation, one that

allocates consumptions to everyone for each realization of the state, and which implicitly agreeswith the bilateral norm for every linked pair

With this setup as background, the paper then studies the stability of insurance networks,explicitly recognizing the possibility that the lack of commitment may destabilize insurance ar-rangements Thus a consumption allocation cannot only be consistent (with the underlying

norm); it must also be self-enforcing.

But precisely what does self-enforcement entail? In the “group-based” insurance paradigm,

a natural supposition is that a deviating individual is thereafter excluded from the group, andthat is what the bulk of the literature assumes Yet if arrangements are fundamentally bilateral,

this sort of exclusion needs to be looked at afresh If A deviates from some arrangement with

B, we take it as reasonable that B refuses to engage in future dealings with A.3 The payoff

consequences of this refusal may be taken to be the weakest punishment for A’s misbehavior But the punishment may conceivably be stronger: B might “complain” to third parties If such parties are linked directly to A they, too, might break their links (such breakage would be

sustained by the usual repeated-game style construction that zero interaction always constitutes

an equilibrium) To go further, third parties might complain to fourth parties, who in turn

might break with A if they are directly linked, and so on Such complaints will travel along a

“communication network” which in principle could be different from the network determiningdirect transfers, but in this paper we take the two networks to be the same.4 If all agents areindirectly connected in this way, then the limiting case in which all news is passed on — andcorresponding action taken — is the one of full exclusion typically assumed in the literature Wepropose to examine the intermediate cases

Our analysis highlights two forces in the relation between the architecture of the network and the stability of insurance schemes: an informational effect that determines the capacity of the network to punish deviants, and a transit or bottleneck effect that arises from the restriction that

transfers must only take place between linked agents

We first study the informational effect To rule out the short-term features we deliberatelyfocus on discount factors close to unity The first principal result (Proposition 3) of the paperprovides a full characterization of those insurance networks that satisfy the self-enforcementconstraint for different “levels” of communication By “level” we refer to the number of rounds

q of communication (and consequent retribution) that occur following a deviation: for instance,

if the immediate victim talks to no one else, q = 0, if she talks to her friends who talk to no one else, then q = 1, and so on For any such q, we provide a characterization of those network architectures that are stable under the class of monotone insurance norms, those in which the

addition of new individuals to a connected component by linking them to one member increasesthat member’s payoff The characterization involves a particular property of networks As an

implication, for any q, typically both thinly and thickly connected networks are most conducive

to stability; intermediate degrees of connection are usually unstable

To obtain an intuitive feel for this implication, imagine that q is small Nevertheless, if

the network is very thin the miscreant may still be effectively cut off (and thereby adequatelypunished) even though the accounts of his deviation do not echo fully through the network (he

3 Of course, issues of renegotiation might motivate a reexamination of even this assumption, but we do not do that here.

4 Certainly, it is perfectly reasonable to suppose that the communication network is a superset of the transfers network So the assumption — that we make to take a first tractable step in this area — is that all pairs who can talk can also make transfers.

was tenuously connected anyway) On the other hand, if the network is fully connected a single

round of complaints to third parties is also enough to punish the miscreant, because there will

be many such “third parties” and they will all be connected to him It is precisely in networks

of intermediate density that the deviant may be able to escape adequate punishment

This suggest a U-shaped relationship between network density and stability for intermediate

level of communication To be sure, this is not a one-to-one relationship as networks of verydifferent architecture have the same average density However, by simulating many networks

of a given density and assessing their stability, we can illustrate this U -shaped relationship.

Similarly, we show a positive relationship between clustering and stability This is particularlyuseful as the density and clustering coefficient (concepts that we will define precisely later) arebasic characteristics of networks used in the social network literature (Wasserman and Faust(1994))

Next, we study the transit or bottleneck effect, and so consider discount factors which are notclose to unity Assessing the stability of mutual insurance schemes in such contexts is a difficulttask We do so assuming a specific risk-sharing norm: equal sharing, in which all pairs of agentsdivide their income (net of third-party obligations but including third-party transfers) equally atevery state

As transfers can only flow along links in the network, the consistent transfer scheme associatedwith such a norm may effectively require excessive reliance on a particular “bottleneck” agent,which in turn increases her short term incentive to deviate from the scheme If we abstract fromthe subtleties of the information effect by assuming that all deviants are punished by full exclusion

(say q is large), then this effect becomes particularly clear, as the post-deviation continuation

value for every agent is the same In this situation, one can identify — for any network —

the “bottleneck” agent by simply looking at the maximal short-term incentive to deviate The

enforcement constraint faced by this bottleneck agent defines the stability of the entire network In

Proposition 4, we isolate a class of “decomposable” networks (which includes all stars and lines)for which the bottleneck effect is identical, and hence stability conditions are identical Moreover,

we show that decomposable networks are the networks for which the bottleneck effects are themost acute The addition of new links can only relax the bottleneck effect, as new links can beused to reroute transfers at every state It follows that adding links only improves the stability

of the network, and that the complete network is stable for lower values of the discount factorthan any other network

However, this finding is for the case of strong punishments, in which all agents are punished byfull exclusion For weaker punishment schemes, we show that a higher density in a network has

an ambiguous effect On the one hand, it reduces the bottleneck effects, thereby helping stability,but, as seen earlier, it also reduces the potential punishment a deviant would suffer which hurtsstability

We believe that this paper represents a first step in the study of self-enforcing insuranceschemes in networks In taking this step, we combine methods from the basic theory of repeatedgames, which are commonly used for models of informal insurance, with the more recent theories

of networks It appears that this combination does yield some new insights, principal amongthem being our characterization of stable networks However, it is only fair to add that we buythese insights at a price For instance, it would be of great interest to study the case in whichthe aggregate of third-party transfers is not observable This would introduce an entirely newset of incentive constraints, and is beyond the scope of the present exercise

Qualifications notwithstanding, our findings contribute to a recent and growing literature

on the influence of network structures in economics See for instance, Calv´o-Armengol and

Jackson (2004) on labor markets, Goyal and Joshi (2003) on networks of cost-reducing alliances,Bramoull´e and Kranton (2004) on public goods, Tesfatsion (1997, 1998) and Weisbuch, Kirmanand Herreiner (2000) on trading networks, Fafchamps and Lund (1997) on insurance, Conley andUdry (2002), Chatterjee and Xu (2004) and Bandiera and Rasul (2002) on technology adoption,and Kranton and Minehart (2000, 2001) and Wang and Watts (2002) on buyer-seller networks.

2 Transfer Norms in Insurance Networks2.1 Endowments and Preferences We consider a community of individuals occupying dif-

ferent positions in a social network (see below) At each date, a state of nature θ (with probability

p(θ)) is drawn from some finite set Θ The state determines a strictly positive endowment y i

for agent i Denote by y(θ) the vector of income realizations for all agents Assume that every

possible inter-individual combination of (a finite set of) outputs has strictly positive probability.[This condition guarantees, in particular, that outputs are not perfectly correlated.]

Agent i is endowed with a smooth, increasing and strictly concave von Neumann-Morgenstern utility u i defined over consumption, and a discount factor δ i ∈ (0, 1) Individual consumption will

not generally equal individual income as agents will make transfers to one another However, weassume that the good is perishable and that the community as a whole has no access to outsidecredit, so aggregate consumption cannot exceed aggregate income at any date

2.2 Networks Agents interact in a social network Formally, this is a graph g — a collection

of pairs of agents — with the interpretation that the pair ij belongs to g if they are directly

linked In this paper, a bilateral link is a given: it comes from two individuals getting to knoweach other for reasons exogenous to the model While such links may be destroyed (for instance,due to an unkept promise), no new links can be created

Note that two individuals are connected in a network if they are directly or indirectly linked, and the components of a network are the largest subsets of connected individuals For any component h of a graph g, we denote by N (h) the set of agents in component h.

For our purposes, a link between i and j means two things First, it means that i and j can

make transfers to each other Second, it is a possible avenue for the transmission of information(more on this below)

2.3 Bilateral Norms In sharp contrast to existing literature, we take a decentralized view

of insurance Any two linked individuals may insure each other This implies some degree ofinsurance for larger groups, but no deliberate scheme exists for such groups To be sure, transfersfrom or to an individual must take into account what her partner is likely to receive from (orgive to) third parties In many situations this is easier said than done Such transfers may not

be verifiable, and in any case all transfers are made simultaneously As a first approximation weassume that for every linked pair, third-party transfers are verifiable ex post, and that the values

of such transfers inform the bilateral dealings of the pair in a way made precise below

In short, for a linked pair ij, state-contingent income vectors (y iand yj) as well as third-partytransfers by each agent (zi and zj) are observed and conditioned upon.5 These latter variablesare endogenous; their exact form will be pinned down in society-wide equilibrium

5Clearly, the third party obligations of agents i and j depend on the specific pair of agents considered Because

we focus on one specific pair of agents, we do not need to take this dependency explicitly into account in our notation.

For instance, consider a three person network in which B is linked to both A and C Use the notation x ij to denote the transfer, positive or negative, from j to i Figure 1 illustrates transfers that equate consumptions in a state in which B and C’s income is 1 while A’s income is 0.

Figure 1 Bilateral Transfer Scheme

A bilateral (insurance) norm is a specification of consumptions for every linked pair of viduals, as a function of various observables Such observables include individual identities i and

indi-j and, of course, the realizations y i , y j , z i and z j So a bilateral norm is just a function b such

that

(c i , c j ) = b(i, j, y i , y j , z i , z j)for every vector of realizations, subject to the constraint

c i + c j = y i + z i + y j + z j

Some norms could be derived from bilateral welfare functions Such a welfare function woulddepend on state-contingent consumptions ci and cj of the two agents, but could also depend onother variables, such as the ambient network component and the identity of the agents Each pair

— viewed as a social entity — by maximizing this function would generate a bilateral norm Forinstance, if the bilateral welfare function consists of the sum of the utilities of the individuals, the

resulting bilateral norm is equal sharing: the bilateral norm simply divides all available resources

among the linked pair

But bilateral norms also include a large class of sharing rules which are not easily amenable

to a welfarist interpretation For instance, suppose that each individual i has full, unqualified access to some fraction α iof her income, and must only share the rest of her resources using, say,

an even split The resulting transfer norm would then look like this:

c i = α i y i+1

2[X

k=i,j (1 − α k )y k + z k]

c j = α j y j+1

2[X

k=i,j (1 − α k )y k + z k]

When α i > 0 for all i, we call these, norms with a private domain.

A bilateral norm aggregates third-party obligations if the consumption of each individual pends on z i and z j through their sum z i + z j alone and is continuously increasing in this variable.Bilateral norms that aggregate third-party obligations can still be asymmetric (i.e., depend on

de-the index i and j), and de-they can also prescribe consumptions that are dependent on individual

incomes in a variety of ways Naturally, there is a large class of norms that satisfy this restriction.For instance, the equal sharing norm and the norms with a private domain do aggregate third-party obligations

Notice that a bilateral norm equivalently prescribes bilateral transfers for every linked pair

and every realization

2.4 Consistent Consumption Allocations Recall that third-party transfers are endogenousfunctions of the state Put another way, the society-wide operation of a bilateral norm not onlygenerates transfers across every linked pair; it also generates the third-party receipts or debtsthat the pair takes as “given” Cutting through the implied circularity, a bilateral norm will

yield a consumption allocation c(θ) for everyone in the network, as a function of the realized state θ Call this a consumption allocation consistent with the norm, or a consistent consumption

allocation for short Obviously, with all the interpersonal interactions in the network, there may

be more than one consumption allocation consistent with any given bilateral norm The followingproposition describes when this cannot happen, so that a unique prediction is obtained

Proposition 1 Suppose that a bilateral norm aggregates third-party obligations Then there is

at most one consumption allocation consistent with that norm.

Proof Suppose the assertion is false Then there are two consistent allocations — c(θ)

and c0 (θ) — and a state θ such that the induced vectors of consumptions across individuals in that state are distinct Then there must be some linked pair ij such that c i (θ) ≤ c 0

i (θ) and

c j (θ) > c 0

j (θ) But then at least one consumption is not strictly increasing in the sum of

third-party obligations, a contradiction

Recall the equal-sharing norm, in which every bilateral transfer is chosen to equalize tion across a linked pair It is easy to see that there is a unique consistent scheme associated withthe equal-sharing norm, which entails “global” equal sharing of total output in any connectedcomponent of the network

consump-The equal-sharing norm, apart from its intrinsic interest, has the feature that there is some

“multilateral norm” with which it is consistent; in this case, multilateral equal sharing Bilateraltransfer norms that allocate to each person a weighted share of consumption (depending perhaps

on that person’s identity or her income realization) also have this feature provided that the relative

weights for every pair {ij} equal the relative weights arrived at “indirectly” by compounding relative weights along any other path joining i to j It is also possible to generate bilateral schemes by maximizing welfare functions that depend (for every linked pair ij) on c iand cj Suchfunctions may also depend on the ambient network component and the identity of the agents, asalso their income realizations, particularly if these serve as proxies for “outside options”

It is also easy to find the consistent consumption allocation associated with any norm with a

private domain For a given income realization, any individual i in a component d of size n will

As a particular instance, take the norm in which all individuals keep half their income and share

the remainder of their resources (α i = 1

2 for all i) Consider a three person network in which

B is linked to both A and C Figure 2 illustrates the transfers associated with the consistent

allocation (1/3, 2/3, 2/3) in a state in which B and C’s income is 1 while A’s income is 0.

Now what about existence? Unfortunately, bilateral norms could be “incompatible enough”

so that a consistent consumption allocation associated with that norm simply fails to exist

As an example, suppose there are three agents connected to each other in a circle Assumethat players 1 and 2 have a social norm that involves giving player 2 two-thirds of their jointendowment Likewise, players 2 and 3 wish to give player 3 two-thirds of their joint endowment,

Figure 2 Consistent Allocation.

and a symmetric circle is completed by players 3 and 1 Obviously, there is an incompatibilityhere, and it manifests itself in the nonexistence of a consistent consumption allocation.6

One way of seeing this incompatibility is to “follow” a natural fixed point mapping that wouldgenerate a consistent consumption allocation, were one to exist If no consistent consumptionallocation were to exist, such a mapping would prescribe larger and larger transfers as “bestresponses” in an ever-increasing spiral If one proscribes unbounded transfers by assumption,

the problem goes away We record this as a proposition, for it tells us that there are no other

existence problems except for the one just described

Proposition 2 Suppose that for every linked pair ij, the bilateral transfer norm is continuous

in z i and z j , and that, in any state, the prescribed transfers cannot exceed some exogenous upper bound (say, the total output produced in society in that state) Suppose, moreover, that the norm never prescribes positive transfers from an individual with non-positive consumption to another with positive consumption.

Then a consistent consumption allocation exists, and exhibits positive consumption for every individual at every state.

We relegate the proof to the appendix

Notice that every consistent consumption allocation c also implies an associated transfer

scheme x: a collection of payments x ij (θ) from j to i (positive or negative) for every linked pair

ij and every state θ The third party obligations are then defined by z i (θ) =Pk6=j:ik∈g x ik (θ),

z j (θ) = Pk6=i:jk∈g x jk (θ) Observe that there may be several transfer schemes associated with

the same consumption scheme: the uniqueness result of Proposition 1 does not apply to transfers

We complete this section with a discussion of monotone norms

2.5 Monotone Norms Say that a bilateral norm is “monotone” if whenever more individuals

are brought into a connected network by being connected to any particular individual, that

individual’s payoff increases Intuitively, more individuals create better insurance possibilities,and a monotone norm should give some of the extra benefits to the individual serving as a

“bridge”

Formally, suppose that g and g 0 , with g ⊆ g 0 , are two connected components such that N (g) ⊂

N (g 0 ) Suppose, moreover, that jk ∈ g 0 only if jk ∈ g for all j, k 6= i Then, say that a norm

is monotone if for every pair of associated consistent consumption allocations, one for g and one for g 0 , the expected payoff to i under g 0 is higher than that under g.

6 The incompatibility does not arise from the weights alone: the linkage structure matters as well For instance, there is no existence problem if the three players are connected “in a line”.

Notice that monotonicity embodies more than a purely normative definition; it requires that

a consistent solution not “misbehave” as we move across networks.7

It is easily seen that when agents are symmetric the equal division norm is monotone It would

be interesting to describe the full class of norms which satisfies monotonicity, though we do notknow the answer to this question

3 Enforcement Constraints and StabilityWhile a bilateral norm, as defined by us, comes from a fairly general class, it is time to

emphasize a particular feature (already discussed in the Introduction) These norms are largely

“normative” in that they take little or no account of self-enforcement constraints But this isn’t

to say that such constraints do not exist Each individual may recognize that as a social being

she is constrained to abide by the transfer norm in her dealings with j, provided that she wants

to maintain those dealings But she may not want to maintain them It may be that (in some

states) the transfer she is called upon to make outweighs the future benefits of maintaining a

relationship with j under the bilateral norm If that is the case, something must give, either the norm or the ij link Our paper takes the point of view that the norm is more durable than the

link, and that the link will ultimately fail.8

In a network setting, an agent could choose to renege on some (or all) transfers that she

is required to make under a particular bilateral norm In line with the bulk of the literature

on risk-sharing without commitment (see, e.g., Coate and Ravallion (1993), Fafchamps (1996),Ligon, Thomas and Worrall (2001) and Genicot and Ray (2003, 2005)), individuals who are the

direct victims of a deviation are presumed to impose sanctions on the deviant thereafter by not

interacting with them

This much may be clear, but nevertheless the extent of the punishment imposed on a deviantremains ambiguous What about the rest of society, who were not directly harmed by the deviant?

Do they, too, sever links with the deviant?

The answer to this question depends in part on what we are willing to assume about the extent

of information flow in the society In turn, this forces us to ask the question of just what thenetwork links precisely mean They certainly limit physical transfers, but do they also limit theflow of information? One possible interpretation is that the network represents a set of physicalconduits and physical conduits alone, while information flows freely across all participants and isnot constrained in any way by the architecture of the network In this case the following notion

of a punishment may be appropriate:

Strong Punishment Following a deviation, every agent severs its direct link (if any) with thedeviant, so that the deviant is thereafter left in autarky

In models of informal insurance in groups with self-enforcement constraints, this is the monly adopted punishment structure But in such scenarios, there are no networks, insurance is

com-7 One possible source of “misbehavior” is nonuniqueness of consistent schemes for a given network While this

in itself is by no means inconsistent with monotonicity, it makes the concept less intuitive.

8 Generally speaking, should we conceive of norms as restricted or unrestricted by incentives? This is an important open question that we do not pretend to address in any satisfactory way Norms may range all the way from the fully idealistic (purely derived from ethical considerations, such as equal-sharing) to the purely pragmatic (wary of all enforcement and participation constraints, with ethical matters only invoked subject to the limits posed by such constraints) In this paper, we take the point of view that norms are not constrained by incentives, but of course we do use such incentive constraints to see if the resulting bilateral norms will or will not survive.

fully multilateral, and the event of a deviation is common knowledge among the group as a whole.

In a situation in which network effects are under explicit consideration, the opposite presumptionmay seem more natural:

Weak Punishment Following a deviation, only those agents who have been directly mistreated

by the deviant sever their links (with the deviant)

In our view, this concept is more appropriate to the case at hand than strong punishment Inthe model that we study, insurance is bilateral, and linked agent pairs know relatively little aboutthe particulars of other dealings (only the aggregate of transfers made to or received from thirdparties).9 So it is entirely consistent to impose the restriction that while directly injured partiesreact, other agents do not, while strong punishments are more appropriate to a multilateralsituation in which there are no restrictions on information flows and no network effects

At the same time, if we take the network structure seriously, not just as a routeway forphysical transfers but also for the flow of information, then we can define “intermediate” layers ofpunishment that are worth investigation in their own right For instance, if I am an injured partyand can communicate with those I am directly linked to, I can tell them about my experience.One might then adopt the equilibrium selection rule that all the individuals I talk to sever directlinks (if any) with the deviant

To be sure, once this door is opened, we might entertain notions in which the news of anindividual’s mistreatment “radiates outwards” over paths of length that exceed a single link,and all those who hear about the news breaks off direct links (if any) with the original deviant.There are many ways to model such a scenario: we take the simplest route by indexing suchpunishments by the length of the required path

Level-q Punishments Following a deviation, all agents who are connected to a victim by a path not exceeding length q (but not via the deviant) sever direct links (if any) with the deviant Figure 3 illustrates this punishment scheme for q = 1 If in a given period individual 1 reneges

on the transfers he owes to 4 and 5, then not only 3 and 4 but also 5 will sever their links to 1

This results in the subgraph g 0

Figure 3 Level-q Punishment.

9 However, for our model to have proper game-theoretic underpinnings — the formalities of which we do not explore here — it will be necessary to assume that the network itself is commonly known.

In this definition, q is to be viewed as a nonnegative integer, so that weak punishment may be thought of as a special case in which q = 0 In this sense, level-q punishments are quite general.10

In Section 6.2, we discuss some other punishment structure

Notice that we effectively treat q as an exogenous parameter to measure the extent of

in-formation flow The reader may ask whether the passing-on of news about deviations may bedetrimental to one’s own interests (see a recent paper by Lippert and Spagnolo (2005) — thoughnot on insurance — for more discussion on this point) This is an important question, but theissue is not critical here as our implicit punishment scheme is sustainable as a Bayesian equilib-rium in a game in which network members send messages along their communication network

(determined q) to report whether the transfer was made or not No individual will make a

posi-tive transfer to another if they do not expect reciprocity It follows that if a deviant believes thatone of his neighbors has been informed of his misbehavior, he will stop making transfers to heranyway Given these beliefs, it is an equilibrium strategy for individuals along the path from thevictim to the neighbor to transmit the information on the deviation Of course, there may beother equilibria in which badly treated partners resume dealings with the deviant or fail to passthe information, but we focus on the worst available punishments Individuals who are directvictims or are informed of a deviation are presumed to transmit the information and imposesanctions according to the punishment scheme

Now, given a level-q punishment structure in place and given a norm, we may define q-stable

networks (Sometimes, when there is no danger of ambiguity, we shall simply use the term

“stable” instead of q-stable.) Consider a community of n individuals and a bilateral norm defined

over all possible pairs of individuals Because deviations result in the severance of links, wedevelop a recursive notion of stability To this end, we begin with the empty graph in whichall individuals live in isolation The expected lifetime utility of an agent living in autarchy(normalized by the discount factor to a per-period equivalent) is

consistent consumption allocation c with expected payoff vector v Fix a transfer scheme x

associated with it Consider any individual i For any realization θ, by abiding with the norm, i

obtains a lifetime (normalized) expected payoff of

In contrast, if i deviates by not honoring commitments to a set of neighbors S, a level-q punishment will set the new graph to g 0 , which is obtained by removing from g all direct links to

i that are from individuals who are no more than q steps distant from some member of S (but the

connecting path should include the original deviant) Thus, the continuation payoff will depend

on two things: the set of players who are her “victims”, and the value of q that determines the

10 Of course, one might conceive of still more general punishment structures in which verifiable information decays — perhaps probabilistically — as it radiates along a path, but we avoid these for the sake of simplicity.

punishment level Formally, the payoff to i of such a deviation is

i in (2) determined? To describe the continuation payoffs following a deviation, we must

adopt a convention that tells us the payoffs that accrue to player i when she finds herself at some network g 0 ⊂ g.

If g 0 is stable, it is to be expected that i will enjoy a payoff of v 0

i , where this is the ith component

of some stable payoff vector for g 0 If g 0 is not stable, the resulting payoff will be presumably

drawn from some stable subnetwork of g 0 itself Two often-used devices to pin down the preciseoutcome in the face of potential multiplicity are “optimistic” and “pessimistic” beliefs (see, e.g.,Greenberg (1990)).11 We assume that if g 0 is not stable then a subnetwork g 00 would form where

g 00 is the or one of the largest stable subnetworks of g 0 to which i belongs In this case, v 0

i is the

ith component of some stable payoff vector for g 00 We do not insist on any particular selection

rule at this conceptual stage, but we must take note of the “baseline” graph that player i induces

on her deviation This depends on two things: the set of players who are her “victims”, and the

value of q that determines the punishment level.

Furthermore, notice that our formulation of the payoff following a deviation implicitly assumesthat an agent’s deviation does not affect the ability of other agents to fulfill their commitments

In other words, we suppose that all other agents continue to conform to the norm after the

deviation of agent i This restriction stands in contrast to the assumption made in the literature

on systemic risks and interbank credit (Allen and Gale (2000), Rochet and Tirole (1996) andFreixas, Parigi and Rochet (2000)) In models of interbank lending, the liquidation of assets held

by bank i in bank j affects bank j’s ability to honor its commitment, and can result in financial

contagion, where the failure of one bank triggers the failure of other banks in the network Weabstract from this important but difficult issue in this paper

Comparing (2) and (1), we may therefore say that a network g is q-stable under a given bilateral

norm if it has a consistent consumption allocation (with expected payoff vector v) and associated

transfer scheme such that for every player i, every state θ and every set of direct neighbors S of

Stability requires that for all possible state realizations, the stipulated transfers be self-enforcing

If q is small, then punishments are “weak”: not many individuals punish the miscreants Indeed, for q = 0 we obtain precisely the notion of weak punishment introduced earlier If q

is large we approximate strong punishment and indeed that is what we get if the network is

connected to begin with In any case, “strong stability” can always be defined by setting v 0

i in

(2) to v ∗

i (∅); no recursion is needed.

Notice that q-stability exploits fully an awareness of repeated interaction between individuals

and is therefore different from the stability concepts in Jackson (2001, 2004) and Baya and Goyal(2000)

11For instance, in the former case, v 0

i would be the maximum value of v idrawn from all stable payoff vectors

drawn from any (stable) subgraph of g 0.

Inequality (3) can be rewritten as

critical for their stability: the transit effect, a short-term effect, and the punishment capacity of

networks, a long term effect The next section focuses on the latter aspect by looking at highvalues of the discount factor In Section 5, we then consider lower discounting values to examinetransit or “bottleneck” effects

4 Stable Networks for High Discount Factors

In this section we characterize the set of q-stable networks for a given bilateral norm The

following notion of a “sparse connectedness”, related to the length of minimal cycles connectingany three agents in a network, will be central to the analysis.12

4.1 Sparseness For any triple of agents (i, j, k) such that (i, j) and (j, k) are connected in the network, compute the length of the smallest cycle connecting i, j and k, `(i, j, k) By convention,

if there is no cycle connecting those three points, we define `(i, j, k) = 2 For any integer q ≥ 0, say that a graph g is q-sparse if all minimal cycles connecting three agents in the network have length smaller or equal to q + 2.

Note that if a network is q-sparse, then it is q 0 -sparse for all q 0 ≥ q Observe also that all

networks which only have trees as their components are 0-sparse — there are no cycles connecting

any three points in a tree — and that this fully describes the set of 0-sparse networks On theother hand, the complete network is 1-sparse, as any three agents are connected by a cycle

of length 3 Actually, we show in the appendix that a network is 1-sparse if and only if itscomponents are clusters connected by bridge-nodes (see Observation 1 in the appendix) Finally,

for a connected network of size n, the graph architecture with the highest index of sparseness

is the circle, where any three agents are connected by a cycle of length n, so that the circle is (n − 2)-sparse As an illustration, Table 1 characterizes the lowest q-sparseness of the different

network architectures for 8 connected agents illustrated in Figure 4

Figure 4 q-sparseness & cycles

12 We are grateful to Anna Bogomolnaia whose comments suggested this definition to us.

graph minimal sparsity

Proposition 3 Suppose that a bilateral norm is monotonic and aggregates third-party

obliga-tions Then a network g is q-stable if and only if it is q-sparse.

It is very important to note that q-sparseness is a purely graph-theoretic property It requires

no knowledge of utility functions, the set of possible income realizations, and the stochastic law

that governs such realizations In contrast, q-stability is a more complex game-theoretic notion.

Not only do we need knowledge of utilities and endowments to define the concept, we need theusual repeated game apparatus of deviations and punishments Therefore, despite the linguistic

similarity in terminology, q-stability and q-sparseness are very distinct concepts, and indeed this

is why our characterization is potentially useful

Proof The proof of necessity uses two steps The first step shows that for the bilateral norms

under review, no punishment can be imposed on a deviant unless it is severe enough to breakdown the connected component of which the deviant is part

Lemma 1 Suppose that a bilateral norm aggregates third-party obligations Then, assuming that

a consistent consumption allocation exists for g, exactly the same allocation is consistent for every connected subnetwork of g.

For a formal proof, see the appendix Bilateral norms that aggregate third-party obligationsmanage to impose the same consumption structure over every set of agents, provided that they

are connected The particular structure of connectedness does not matter For instance, the equal division bilateral norm actually gives rise to global equal division over any set of agents, as

long as they form a connected network

The second main step is a restatement of q-sparseness.

Lemma 2 A graph g is q-sparse if and only if, for every linked pair ij ∈ g, the graph formed by

removing from g the links to i along all paths of length m ≤ q + 1 between i and j has strictly more components than g.

Once again, the proof is postponed to the appendix Now connect Lemmas 1 and 2 Consider

a network and a bilateral norm that aggregates third-party obligations Assume that an agent i

reneges on the transfers she owes to one or more partners Then the immediate victims certainly

sever their links to i, and so do all individuals (connected to i) who are connected to any of them (but not through i!) via a path of length q or less If the graph g is not q-sparse, then, in the resulting subgraph, i will not find herself in a smaller component, by Lemma 2 By Lemma 1, there can be no reduction in continuation utilities following the deviation, so q-stability is not

possible